Properties

Label 460.2.a.e
Level $460$
Weight $2$
Character orbit 460.a
Self dual yes
Analytic conductor $3.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} + ( - \beta + 1) q^{7} + (\beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{5} + ( - \beta + 1) q^{7} + (\beta + 1) q^{9} + 2 q^{11} + (\beta - 2) q^{13} + \beta q^{15} + ( - \beta + 1) q^{17} + 6 q^{19} - 4 q^{21} + q^{23} + q^{25} + ( - \beta + 4) q^{27} + ( - 2 \beta + 3) q^{29} + ( - 4 \beta + 1) q^{31} + 2 \beta q^{33} + ( - \beta + 1) q^{35} + (\beta - 3) q^{37} + ( - \beta + 4) q^{39} + ( - 2 \beta + 1) q^{41} + (\beta + 1) q^{45} - 3 \beta q^{47} + ( - \beta - 2) q^{49} - 4 q^{51} + (\beta - 3) q^{53} + 2 q^{55} + 6 \beta q^{57} + (3 \beta + 1) q^{59} + (2 \beta - 4) q^{61} + ( - \beta - 3) q^{63} + (\beta - 2) q^{65} + (\beta - 7) q^{67} + \beta q^{69} + ( - 2 \beta + 7) q^{71} + ( - \beta - 6) q^{73} + \beta q^{75} + ( - 2 \beta + 2) q^{77} + (2 \beta + 8) q^{79} - 7 q^{81} + (7 \beta - 3) q^{83} + ( - \beta + 1) q^{85} + (\beta - 8) q^{87} + ( - 4 \beta + 8) q^{89} + (2 \beta - 6) q^{91} + ( - 3 \beta - 16) q^{93} + 6 q^{95} + (2 \beta - 10) q^{97} + (2 \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} + 4 q^{11} - 3 q^{13} + q^{15} + q^{17} + 12 q^{19} - 8 q^{21} + 2 q^{23} + 2 q^{25} + 7 q^{27} + 4 q^{29} - 2 q^{31} + 2 q^{33} + q^{35} - 5 q^{37} + 7 q^{39} + 3 q^{45} - 3 q^{47} - 5 q^{49} - 8 q^{51} - 5 q^{53} + 4 q^{55} + 6 q^{57} + 5 q^{59} - 6 q^{61} - 7 q^{63} - 3 q^{65} - 13 q^{67} + q^{69} + 12 q^{71} - 13 q^{73} + q^{75} + 2 q^{77} + 18 q^{79} - 14 q^{81} + q^{83} + q^{85} - 15 q^{87} + 12 q^{89} - 10 q^{91} - 35 q^{93} + 12 q^{95} - 18 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.56155
2.56155
0 −1.56155 0 1.00000 0 2.56155 0 −0.561553 0
1.2 0 2.56155 0 1.00000 0 −1.56155 0 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.a.e 2
3.b odd 2 1 4140.2.a.m 2
4.b odd 2 1 1840.2.a.m 2
5.b even 2 1 2300.2.a.i 2
5.c odd 4 2 2300.2.c.h 4
8.b even 2 1 7360.2.a.bi 2
8.d odd 2 1 7360.2.a.bo 2
20.d odd 2 1 9200.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.a.e 2 1.a even 1 1 trivial
1840.2.a.m 2 4.b odd 2 1
2300.2.a.i 2 5.b even 2 1
2300.2.c.h 4 5.c odd 4 2
4140.2.a.m 2 3.b odd 2 1
7360.2.a.bi 2 8.b even 2 1
7360.2.a.bo 2 8.d odd 2 1
9200.2.a.bv 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(460))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 13 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 67 \) Copy content Toggle raw display
$37$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} - 17 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 38 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 19 \) Copy content Toggle raw display
$73$ \( T^{2} + 13T + 38 \) Copy content Toggle raw display
$79$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} - T - 208 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 32 \) Copy content Toggle raw display
$97$ \( T^{2} + 18T + 64 \) Copy content Toggle raw display
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